hello hello everyone hope you're doing well in our last two lessons we dealt with arrangements or permutations we saw arranging things in a line as well as arranging things around a circle linear and circular permutations and when we dealt with those questions we had to pick things in a particular order when you're running a race of three people you've got three choices for who can get first and then two choices for who can get second and then that leaves one person to finally get third but the order in which you chose them you assign to first and then you assigned a second then you assigned a third there was order built into the question but what we're now going to look at is a technique for counting when we don't care about order yeah we care about the end result not the order in which it happened so suppose we've got a collection of things you know books and tea cups courses that we want to choose and we want to pick some of them and we want to pick some books off the shelf we want different books but maybe we don't care which book we pick first we're gonna read them all we just care about which books we're gonna end up reading this is the idea of selection without replacement okay this is different than say picking topics for an essay where everyone can pick to make a topic on the Vietnam War World War 1 or World War 2 or only some people can and one person's choice does not affect another person's choice now when we're selecting our books off the shelf we can't pick the same book twice once it's picked it's pimped so this is selection without replacement sometimes called unordered selection and we're gonna try and better understand this and how to actually compute and calculate for this sort of accounting problem so all we care about is what is selected at the end we do not care about the order in which we take the books off the shelf the order in which we pick the classes for our schedule and we just care about the end result so here's a nice motivating example we've got Arthur Bella Colin Danielle and Eve we wish to select two of them in order to form a committee for this year's yearbook okay we've got five volunteers Arthur Bella Colin Danielle and Eve but we only need two of them so we need to select them now we can't select Danielle and then select Danielle again and once she's on the committee we got to pick someone else so there's lots of different possibilities could have the committee that has Arthur and Eve and we're gonna use this notation here to help sort of symbolize the committees that were forming so Arthur and Eve maybe maybe we have everybody sort of line up and Arthur and Eve step forward and Bella Colin and Danielle stay at the back or maybe Colin and Danielle are the ones who step forward and volunteer for the yearbook or step forward because we selected them but in the end we've got the Colin and Danielle yearbook committee so we've got lots of possibilities how many how many possible committees could we have how many different possibilities are there so that's our challenge now we can answer this question by making a series of choices we can impose some order on the question artificially by saying well first I'll choose Arthur and then I'll choose Bella five choices for who to pick so I pick Arthur out of the five four people left maybe I pick Bella Arthur then Bella I get the Arthur Bella committee but we have to acknowledge that if I choose Arthur and then Bella that's the same as if I chose Bella and then Arthur I don't care about the order in which they were chosen I care about the end result the Arthur Bella yearbook committee and there isn't a president or vice-president there are no labels going on here so doesn't matter who was picked first there's no way to tell it apart the Arthur Bella versus Bella Arthur committee and it's just who's on the committee at the end that's all we really care about as a result if we try and just say well five choices for the first student times four choices for the next student we're going to over count because we will count to these as separate situations when really they're not so if we experiment with a little bit five students isn't that many and we can get ten different possible committees Arthur Bella Arthur Colin Arthur Danielle Arthur Eve Bella Colin Bella Danielle Bella Eve Colin Danielle Colin Eve Danielle and Eve ten different committees you can try you're not gonna find any others so with five students and we need to select two of them ten possible committees all right so we managed to list out the answer here in a very organized fashion but we still listed it out and we got our answer of ten now we could ask the more general question if I have n students and I need to select K of them for my yearbook committee how many ways can I do this if all I care about is the objects that are selected and not the order and not the order of selection how many ways are there to do this and things to choose from and I need to choose K of them now there is a way we could create a formula for this but it's not a nice everyday formula I mean we'll see it in a moment but it's not the easiest thing to work with and it turns out selection without replacement comes up a lot so what mathematicians decided to do is they decided to create some notation it's a brand new notation and this is pronounced n choose K it's comforted sometimes called selection notations I prefer to call it choose notation you can also call it binomial notation binomial so n choose K is how this is pronounced and what does it mean well it just it's notation that represents exactly what we want the number ways to select K things from a collection of n of them and there is a formula involving factorials that we can use and choose K is n factorial over K factorial times n minus K factorial so for us with our question we could have had 5 choose 2 which we now know is 10 now some notation values are good to have memorized so like 0 choose 0 is equal to 1 1 choose 0 and 1 choose 1 these are both also equal to 1 2 number of ways to choose no things if I got a set of 2 well one way I just don't do anything yeah I've got 2 objects and I want to pick one of them I've got two choices this one or that one and so we can slowly write out and work out some of these using the formula you want 3 choose 2 that would be 3 factorial over 2 factorial over 3 minus 2 factorial which would be 6 over 2 3 minus 2 is 1 so 1 factorial and look at that it is 3 and I do recommend just like you should know some small factorials you should know some nice simple choose notation values usually up to a 4 on the top so now if I've got four things and I want to select two of them there are six ways to do it and as we do more questions we might encounter specific values as I already said 5 choose 2 we saw already is 10 5 students I need to choose two of them and we saw that that answer was 10 so it's good to know these these are arranged in a very particular order here it does represent a triangle or resemble a triangle and if you're familiar with this this is something called Pascal's triangle we're not gonna make terrible use of that but if you've seen it before you might recognize it and if you want to get to know but a little bit better that's something you can Google now in addition to knowing nice small values like 4 choose 2 is 6 it's a good idea to know some nice properties just like when it came to factorials it's nice to know how to rewrite 1 factorial in terms of the next smallest factorial well it's good to know that n to 0 is 1 and this is very helpful yeah we'll see it a lot later on we've got n things there's only one way to choose none of them that's don't do anything just sit there and stare at them and choose one is n because you've got n different possible things to pick from just pick one of them and n choose n is 1 there's one way to choose and objects out of a set of N and that's by choosing all of them so these are nice simple ones there are other ones you can memorize if you really want to if you do more advanced math you'll find that n choose 2 is always n times n minus 1 over 2 but I'm not suggesting that ones that that's one that you have to know but these 3 definitely are good to know another incredibly useful property especially when it comes to answering questions and trying to match your answer to presented answers is that n choose K is equal to n choose n minus K so what would this tell us 5 choose 2 so 5 is N and K is 2 is the same as 5 choose 5 minus 2 which would be 3 and so both of these would be 10 okay this is a very very important result and we can sort of see why it is if we imagine our students we can think of the yearbook committee as the people stepping forward and that makes a group that makes the yearbook committee group but by them stepping forward we have separated all 5 students into two groups actually the yearbook committee and the not yearbooks committee you might think of it as okay so choosing two people is the same as not choosing three people and we're but we're still distinguishing some students and that's how you can see that five choose two should be the same as five choose a three okay so another way to think of this choose notation is the number of ways to break and objects into two different groups the chosen group and the not chosen group so that's the only another approach but what this notation tells us is that you know I'm doing a question I've got all these calculations and I get my answer is eight choose three maybe I look amongst all the multiple-choice answers and I don't see a choose three but off in the corner I see eight choose five it's important remember these are the same thing they mean the same number they're just expressed slightly differently so it's definitely good to have a handle on that switching so you want to know values for you choose notation it's a good idea to know the actual formula it's an even better idea to know what this represents number of ways to choose K things out of N Things or a number of ways to split a group of n objects into two groups one of size K one of size n minus K and it's also a good idea to know these properties yeah they can help you simplify things or just rewrite things if you need a different expression so we're actually going to make use of choose notation a lot not just when we're doing counting problems but even further on in the course is it probably the most common of the notations that we have created for ourselves